Recently, various systems and apparatus for machining workpieces have been developed in which a laser beam constitutes the actual working tool for abrasive cutting, welding and/or machining. Apart from the abrasive cutting and welding of sheet metal, foils or the like, one field of application for correspondingly designed laser machines resides particularly in the machining and shaping of deeper cavities in solid workpieces. This field of application will become ever more important in the future, especially for machining formerly critical materials such as high alloy steels, cutting metals, ceramic materials, composite materials, glasses, etc. A number of publications such as, for example, German Pat. No. 2,443,334, German utility model 8701354.1, Japanese patent application No. 59-47086 and U.S. Pat. No. 4,644,126, disclose machining techniques using laser beams, in which a high energy CO.sub.2 -laser is focussed by means of an optical system and strikes the work together with a machining or auxiliary gas. However, all of these systems are limited to pure laser machining.